Statistical Field Theory for Polar Fluids Bilin Zhuang, and Zhen-Gang Wang

Statistical field theory for polar fluids Bilin Zhuang, and Zhen-Gang Wang

Citation: The Journal of Chemical Physics 149, 124108 (2018); doi: 10.1063/1.5046511 View online: https://doi.org/10.1063/1.5046511 View Table of Contents: http://aip.scitation.org/toc/jcp/149/12 Published by the American Institute of Physics THE JOURNAL OF CHEMICAL PHYSICS 149, 124108 (2018)

Statistical ﬁeld theory for polar ﬂuids Bilin Zhuang1,2 and Zhen-Gang Wang1,a) 1Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA 2Department of Materials Science and Engineering, Institute of High Performance Computing, Singapore 138632, Singapore (Received 28 June 2018; accepted 3 September 2018; published online 26 September 2018)

Using a variational field-theoretic approach, we derive a theory for polar fluids. The theory naturally accounts for the reaction field without resorting to the cavity construct and leads to a simple formula for the dielectric constant in terms of the molecular dipole moment and density. We apply our formula to calculate the dielectric constants of nonpolarizable liquid models for more than a hundred small molecules without using any adjustable parameters. Our formula predicts dielectric constants of these nonpolarizable liquid models more accurately than the Onsager theory and previous field-theoretic dielectric theories, as demonstrated by the closer agreement to the simulation results. The general theory also yields the free energy, which can describe the response of polar fluids under applied electric fields. Published by AIP Publishing. https://doi.org/10.1063/1.5046511

I. INTRODUCTION detailed information about the molecular interaction is often unavailable.17,20 Since then, more sophisticated theories for The dielectric constant is one of the most basic properties the dielectric constant have been developed based on per- of fluids. It is fundamental to many aspects of science and turbative expansions,21–23 cluster expansions,24–28 integral- technology, including behaviors of biological molecules,1,2 equation theories,29–34 and by considering the specific shapes mechanisms of chemical reactions,3–5 fabrication of new mate- of molecules35,36 or intermolecular bonding.37 However, the rials,6–8 solubility of polar and nonpolar species,9,10 and devel- Debye equation and the Onsager equation remain popular due opment of energy storage devices.11 Accurate determination of to their simplicity. the dielectric constant from molecular parameters, especially Despite the many attempts, there remains a need for a sim- in the liquid phase, remains a challenge, whether theoretically ple theory for calculating dielectric constants of fluids based or through computer simulations.11 on readily available molecular parameters. Methods based on Many attempts have been made.12,13 In 1912, Debye perturbative expansions are unsatisfactory because the series derived an equation relating the dielectric constant to the often do not converge with increasing number of terms.38 molecular permanent dipole moment and the isotropic polar- Cluster expansion methods and integral-equation methods are izability. As Debye assumed each dipole independently inter- able to relate dielectric constants to molecular properties, but acts with the applied field, his theory did not take into the required five-dimensional correlation functions (three for account the intermolecular dipole-dipole correlations; there- positions and two for orientations) are very cumbersome to fore, the Debye theory is only applicable for gas-phase sys- evaluate.28,39 tems in the dilute limit.14,15 In 1936, Onsager developed In recent years, progress in field-theoretic statistical- a theory by considering a point dipole in a cavity that is mechanical methods has provided an avenue for the embedded in a homogeneous dielectric continuum. This point development of new and simple expressions for dielectric dipole induces polarization in the surrounding medium, which constants. An additional important advantage of the field- gives rise to a reaction field that in turn influences the theoretic approach is that it can be conveniently integrated into dipole alignment in the cavity.16 The Onsager theory yields a wide range of platforms for studying soft-matter and poly- a much improved prediction for liquid dielectric constants meric systems where the molecules are modeled with the same than the Debye theory, as validated by experimental mea- approach.40–42 One of the simpler field-theoretic methods is surements.17 Shortly after, Kirkwood extended Onsager’s the- the self-consistent-field theory (SCFT), where one approxi- ory by introducing a g-factor to account for the short-range mates the partition function using the saddle-point approxi- orientational correlation between the molecules.18 While mation. Several studies have applied this approach43–47 and Kirkwood’s formulation is especially relevant for determin- the results are comparable to the predictions of the Debye the- ing dielectric constants with computer simulation,19 it is ory. Beyond the saddle-point approximation, Levy, Andelman, often difficult to evaluate the g-factor analytically because and Orland derived an expression for the dielectric constant by considering the one-loop expansion of the free energy.48,49 More recently, Martin et al. derived the dielectric constant a)[email protected] by pursuing a simple gaussian field approximation to the

0021-9606/2018/149(12)/124108/13/$30.00 149, 124108-1 Published by AIP Publishing. 124108-2 B. Zhuang and Z.-G. Wang J. Chem. Phys. 149, 124108 (2018)

field-theoretic Hamiltonian.50 However, the numerical per- molecule has a permanent dipole moment of magnitudeµ ¯. formance of the predictions by these field-theoretic theories The microscopic state of the system can be specified by the is less than satisfactory when compared to the simulation or set of positions and dipole vectors of all molecules {(ri, µi): experimental data. In this work, we develop a field-theoretic i = 1, 2, ..., N}, where ri and µi denote, respectively, the posi- theory for calculating dielectric constants of pure liquids using tion and the dipole vector (where |µi | = µ¯) of the ith molecule. a variational approach,51,52 with the goal to improve the over- The electrostatic interaction energy U of the system is the sum all accuracy in predicting dielectric constants using only a few of all pairwise dipole-dipole interactions and the interaction readily available parameters. energy with the applied field, The dipolar renormalized gaussian fluctuation (DRGF) N N N theory presented in this work is formulated using a field-based 1 X X X U = µ T(r − r )µ − µ · E (r ), (1) gaussian reference action that involves an interaction tensor 2 i i j j i 0 i i=1 j=1 i=1 and the average of the fluctuating field as the variational param- i,j eters. The self-consistency condition on the interaction tensor and the average field naturally accounts for the effects of the where T(r) = −∇∇(1/4πε0|r|) is the dipole-dipole interaction reaction field. While in the current work we consider nonpo- tensor, with ε0 being the vacuum permittivity. larizable fluids for simplicity, effects of polarizability can be Mathematically, improper integrals involving T(r) are included in a straightforward way and we reserve the effort nonunique and lead to a problem of “conditional convergence.” 54,55 for an upcoming paper. The resulting theory consists of sim- However, the problem is well understood and a consistent ple analytical expressions for the free energy and the dielectric interpretation is obtained by representing T(r) in the following 56 constant. In particular, the dielectric constant of a liquid can form: be calculated based on the permanent dipole moment and the 1 1 number concentration of the molecules, both being readily T(r) = −H(|r| − η)∇∇ + 1δ(r) 4πε0|r| 3ε0 available parameters for many systems, without the need for ! 1 3rr 1 any fitting parameters. We apply the DRGF theory to calculate = H(|r| − η) 1 − + 1δ(r), (2) 3 2 the dielectric constants of nonpolarizable molecular models 4πε0|r| " |r| # 3ε0 of a wide range of liquids, and we find the results from the where 1 is the unit dyadic, δ(r) is the Dirac delta function, DRGF theory provide better agreement with simulation data and H(x) is the Heaviside step function. η is a regularization than previous field-theoretic approaches.43–50 The line of best- parameter which will eventually be taken to be 0 in the final fit through our predicted dielectric constants vs. simulation calculations. Since both terms in T(r) are well defined, it is data has a slope very close to 1 on a log-log plot, which is now straightforward to compute the Fourier transform of T(r) better than the Onsager theory. and the result is39,56 The rest of this article is organized as follows. In Sec.II, we describe the model and formulate the exact partition func- kk T(k) = , (3) 2 tion of the system in the field-theoretic representation. Then, ε0k we introduce a gaussian reference action and apply the vari- H ational approach based on the Gibbs-Feynman-Bogoliubov where a tilde above a quantity denotes the Fourier trans- ˜ −ik·r 2 variational principle to derive a general theory for the free form f (k) = ∫ dr f (r)e . Noting that kk/k is a projection energy of dipolar liquids. We then consider a weak applied operator, we have the following relation: field in the linear response regime to obtain an expression n ε T(k) = ε T(k), (4) for the dielectric constant of a liquid. In Sec. III, we apply 0 0 our theory to calculate the dielectric constants for the GAFF which turns out to be usefulH in the algebraicH manipulations. In (generalized Amber force field) and the OPLS/AA (all-atom addition, T(k = 0) = 1/3ε0 follows from Eq. (2). optimized potential for liquid simulations) molecular mod- For convenience, we develop the statistical mechanics for els for an extensive set of over a hundred molecular liquids the polarH liquid in a grand canonical ensemble of the system that have been studied in benchmark simulations by Caleman under chemical potential µ, inverse temperature β = 1/kBT, 53 et al. For comparison, we also compute the dielectric con- and volume V. The grand partition function of the system is stants with the Onsager equation and previous field-theoretic then given by methods (based on the saddle-point approximation, the one- ∞ X eβµN loop expansion, and the gaussian field approximation). Finally, Ξ = Z(N), (5) N! in Sec.IV, we summarize the key features of the DRGF theory N=0 and offer some concluding remarks. where Z(N) is the canonical partition function for a system of N particles, in which the Boltzmann factor is integrated over II. FIELD-THEORETIC VARIATIONAL THEORY the N-particle configuration space as FOR POLAR LIQUIDS 1 Y dΩ A. The model Z(N) = dr i e−βU . (6) Λ3N i 4π i We consider a system of N polar molecules at uniform density ρ in the presence of an applied field E0(r). The In the above expression, Λ is the thermal de Broglie wave- polar molecules are modeled to be nonpolarizable and each length and the integral over the dipole moment is carried 124108-3 B. Zhuang and Z.-G. Wang J. Chem. Phys. 149, 124108 (2018)

out over the solid angle Ωi of the permanent dipole. The full In the above expression, we have defined T = βT and βµ 3 interaction energy in general includes nonelectrostatic interac- E0 = βE0 for simplicity of notation. λ = e /(4πΛ ) is the tions between the particles. These interactions can be included fugacity of the molecule scaled by the inverse temperature. explicitly in the field theoretical model,42 or more simply by The last term in L, which arises from the single-molecule par- imposing the condition of local incompressibility, in a coarse- tition function in the auxiliary field iG, makes the integral grained description of the excluded volume effects.57 Since for Ξ non-gaussian. In Sec.IIB, we introduce an approxi- these nonelectrostatic interactions do not introduce significant mation scheme for the field-based partition function and the fluctuation effects in a homogeneous bulk liquid, they may be grand potential, from which we derive other thermodynamic treated at the mean-field level. These mean-field contributions properties. can be readily included in the overall free energy. In the present work, we do not explicitly account for these nonelectrostatic interactions in the formulation of the theory in order to leave B. The variational approximation freedom for the different forms and treatments of these interactions. From Eq. (5), the grand potential W of the system is The non-gaussian nature of the exact field-theoretic Ξ given by partition function makes it impossible to evaluate the func- βW = − ln Ξ. (7) tional integrals exactly. A popular approximation is the self- consistent-field theory, in which one simply approximates To move toward a continuum field description, we intro- the partition function by taking the saddle-point value of ˆ duce the microscopic polarization P(r), the field-theoretic action with respect to the functional argu- XN ments. However, the self-consistent-field approach does not ˆ P(r) = µiδ(r − ri), (8) take into account fluctuations in the polarization and the i=1 auxiliary fluctuating field and as a result cannot capture the so the energy U may now be rewritten as reaction field effects. In this section, we apply a variational approach to provide an approximate treatment of the non- 1 U = dr dr0 Pˆ (r)T(r − r0)Pˆ (r0)− dr Pˆ (r)·E (r). (9) linear fluctuation effects that are responsible for the reaction 2 0 field. The first term in Eq. (9) now includes self-interaction of the We begin by introducing a reference gaussian action L0 1 form 2 µiT(ri − ri)µi which is divergent for point dipoles. Such as an approximate description for our system, a divergence can be avoided by introducing a distribution of finite spatial width as in Ref. 51 or by imposing a momentum 1 L [P, G] = dr dr0P(r)T(r − r0)P(r0) + i P(r) · G(r) cutoff in the Fourier integral; we choose the latter. 0 2 The pairwise particle-particle interactions entail enor- 1 − P(r) · E (r) − dr dr0[iG(r) − F(r)] mous degrees of freedom in the configuration space, mak- 0 2 ing evaluation of the configurational integral intractable. To × −1 − 0 0 − 0 decouple the pairwise interactions, we employ field-theoretic A (r r )[iG(r ) F(r )], (12) techniques to transform the pairwise interactions into interactions between each particle and a fluctuating field.42 Here, where the field-interaction tensor A(r) and the average fluc- since the dipole-dipole interaction tensor T(r) does not have tuating field F are the variational parameters to be deter- −1 an inverse operator, we perform the transformation using the mined below. The inverse of the operator A is defined via 0 0 −1 0 00 00 Faddeev-Popov method.40,42,58 The resulting partition func- ∫ dr A(r − r )A (r − r ) = 1δ(r − r ). In the absence of tion becomes one involving quadratic interactions between the external field E0, the field-interaction tensor A is isotropic. the field variables and a one-body interaction between the Although it is possible to account for the anisotropy of A for particles and the auxiliary field. Upon integration over the arbitrarily large E0, in this work, we are concerned with the molecular degrees of freedom, the resulting grand partition weak field limits. Therefore, for simplicity of the theory, A(r) function is written as functional integrals over the polariza- is assumed to be isotropic such that A(r) = a(r)1, where tion and the auxiliary fluctuating field. The transformation a(r) now is the scalar variational parameter that describes the is formally exact; we defer the details of the transformation field-interaction strength. to Appendix A. Here, we simply write down the resulting The exact field-theoretic action L and the reference gaus- field-based grand partition function as sian action L0 only differ in their respective last terms. Essentially, the interaction between a single molecule and the Ξ = DP DG e−L[P,G], (10) auxiliary field in the exact action L is replaced by a quadratic interaction of the auxiliary field in the approximate action where the field-theoretic action L is given by L0. This quadratic interaction makes the overall form of the partition function gaussian, with renormalized variational 1 0 0 0 L[P, G] = dr dr P(r)T(r − r )P(r ) parameters A(r) and F that are determined self-consistently. 2 Therefore, we shall refer to our theory as the dipolar renor- + i P(r) · G(r) − P(r) · E0(r) malized gaussian fluctuation (DRGF) theory. Using the reference action, we can obtain an approx- − λ dr dΩ eiµ·G(r). (11) imation of W based on the Gibbs-Feynman-Bogoliubov inequality59 124108-4 B. Zhuang and Z.-G. Wang J. Chem. Phys. 149, 124108 (2018)

βW ≤ − ln Ξ0 + hL − L0i0, (13) ρV = λ dr dΩ exp[−f (µ, r)]. (21) where Ξ0 is the reference partition function given by The effective single-particle potential f (µ, r) in Eq. (17) Ξ = DP DG e−L0[P,G] (14) describes how a dipole interacts with the polar environment 0 and the applied field. The first term in f (µ, r) describes the self- and hOi0 is the average of an observable O evaluated in the interaction of a dipole in the effective field, with the effective reference ensemble, i.e., self-interaction strength quantified by the tensor TR. Physi- 1 cally, this effective self-interaction includes the effects of the −L0[P,G] hOi0 = DP DG O[P, G]e . (15) reaction field in condensed polar liquids—as a dipole exerts a Ξ0 field on its surroundings, the surroundings exert a field back Since we will approximate βW by the stationary value of on the dipole itself. From Eq. (20a), in the dilute limit (λ → 0), 60,79 the right-hand side in Eq. (13), for economy of notation, a˜(k) → ∞, so the reaction field vanishes and the self-energy we will use W to denote the variational free energy, with 1 term becomes 2 µT(0)µ, which is precisely the self-energy the understanding that only its stationary value corresponds of an isolated dipole moment. The second term in f (µ, r) (approximately) to the equilibrium free energy. The evalua- describes the interaction of a dipole with an effective local tion of W involves performing a series of gaussian functional directing field γ(r) on the dipole. integrals, the details of which we defer to Appendix B. The The constitutive relations Eqs. (20a) and (20b) are to be resulting expression for W is solved simultaneously for the variational parameters a and F, ! 1 dk ε a˜(k) β and these relations also allow us to infer physics of polar flu- βW = − V ln 0 + 3 ids. From Eq. (20a), we observe that the stationary value of 2 (2π) " β + ε0a˜(k) β + ε0a˜(k) # !2 the field-interaction parameter a is related to the averaged self- 1 dk ε − 0 E (k) − F(k) interaction under the effective single-particle potential. As we 2 π 3 β + ε a˜(k) 0 (2 ) 0 f g have alluded to earlier, this self-interaction is a manifestation H H of the reaction field; see Sec. II C 3 for further discussion. × T(k) E (−k) − F(−k) − λ dr dΩ e−f (µ,r). 0 Then, from Eq. (20b), we see that the average auxiliary field f g H H H (16) F is a combination of the applied field E0 and the angu- larly averaged dipole moment in the effective single-particle The last term in βW can be regarded as the renormal- potential. This combination suggests that the auxiliary field ized single-particle partition function, with f (µ, r) being the fluctuates around an averaged value that is dependent on both effective single-particle potential given by the applied field and the electric field generated by the dipolar 1 material. f (µ, r) = µ · T · µ − γ(r) · µ, (17) 2 R The variational grand potential W in Eq. (16) is the cen- terpiece in our statistical field theory, from which all other where thermodynamic properties can be derived. By finding the val- dk ε a˜(k) T = 0 T(k) ues of a and F that allows W to take its stationary value, R 3 (2π) β + ε0a˜(k) we obtain an approximate theory that naturally and self- dk Hβ consistently accounts for the effects of the reaction field in = T(0) − T(k) (18) 3 (2π) β + ε0a˜(k) a polar fluid. In Sec.IIC, we apply the theory to a polar fluid H in a weak applied field and explore its various properties. T dk T with (0) = ∫ (2π)3 (k) and H ε0 γ˜ (k) = E0(k) − T(k)[E0(k) − F(k)]. (19) C. The linear response regime β + ε0a˜(k) 1. The solution of variational parameters We determineH the stationaryH conditionH forH the free energy Eq. (16) by setting ∂W/∂a˜(k) = 0 and ∂W/∂F(k) = 0. Upon When the applied field is weak, the polarization of the fluid minor simplification of the functional derivatives (with details responds to the applied field linearly. In this linear response provided in Appendix C), we obtain the followingH two con- regime, we may solve the constitutive relations, Eqs. (20a) and stitutive relations, which are to be solved simultaneously to (20b), to first order in the applied field. At this order, Eq. (20a) determine the variational parameters: reduces to

βV βV − 1 µ·T ·µ = λ dr dΩ µ · T(k) · µ e−f (µ,r) (20a) = λ dr dΩ µ · T(k) · µ e 2 R (1 + γ(r) · µ) ε0a˜(k) ε0a˜(k) H H − 1 µ·T ·µ and = λ dr dΩ µ · T(k) · µ e 2 R , (22) ! ε 0 E (k) − F(k) = λ dr dΩ µ e−f (µ,r)−ik·r. H β + ε a˜(k) 0 where the second equality results from the integral over an odd 0 f g H H (20b) power of the permanent dipole moment. As such, the solution In addition, the fugacity λ can be related to the ﬂuid density to a˜(k) is isotropic with respect to k. Consequently, the effec- using the statistical mechanics relation ρV = hNi = −∂W/∂µ tive self-interaction tensor TR is also isotropic, and it may be as written as 124108-5 B. Zhuang and Z.-G. Wang J. Chem. Phys. 149, 124108 (2018)

dk ε a˜(k) ∂ βW T = 0 T(k) = T 1, (23) P(k) = −(2π)3 . (30) R 3 R (2π) β + ε0a˜(k) ∂E0(−k) H H where T R is a scalar characterizing the strength of the effective By performing the derivativeH to first order in the self-interaction. applied field, we obtain the following expression for the The isotropic nature of TR results in considerable simpli- polarization: fications in the solution of the constitutive relations since the !2 !2 term 1 µ · T · µ = 1 T µ¯2 is now independent of the direction y ε0 2 2 R 2 R P(k) = T(k) 1 − K(k) E0(k) of µ. The relation between the fluid density and the fugacity y + 1 β H H ! H H 2 is, to the linear order in the applied field, ε0 y ε0 + y 1 − T(k) 1 − K(k) E0(k). (31) − 1 T µ¯ 2 β " y + 1 β # ρ = 4πλe 2 R . (24) H H H Simplification of the above expression with the use of Eq. (4) The constitutive relations Eqs. (20a) and (20b) are now leads to simplified to ! !2 2 β ε0 ε0 y 2 = y (25a) P(k) = yE0(k) − 1 − K(k) T(k)E0(k). ε0a˜(k) β β 1 + y and H H H H H (32) E (k) − F(k) = (1 + y)γ˜ (k), (25b) 0 The expressions for polarization allow us to extract 2 where y = β ρµ¯ H/3ε0 is aH dimensionless parameter character- the dielectric constant of the fluid. To do so, we note izing the strength of dipolar interactions in the absence of the that the electric susceptibility matrix is defined through applied field. P(k) = ε0 χ0E0(k)/β. From Eq. (32), we identify the electric As seen in Eq. (25a), the solution to the variational param- susceptibility matrix as eter a˜(k) is independent of k in the linear response regime. H H H 2 The solution to the other variational parameter, F(k), can be y 2 ε0 χ0(k) = y1 − 1 − K(k) T(k). (33) obtained by substituting the expression for γ˜ (k) in Eq. (19) 1 + y β H H H into Eq. (25b). This substitution leads to Then, the dielectricH constant can be found using the relation39 ! ! ε0 ε0 (ε − 1)(2ε + 1) 1 + y T(k) F(k) = −y 1 − T(k) E0(k) (26) = tr χ (k = 0). (34) β β ε 0 H H H H which gives the solution H Making use of T(k = 0) = ∫ dr T(r) = (β/3ε0)1, we obtain the following simple expression for the dielectric constant: F(k) = K(k)E0(k), (27) H ! (ε − 1)(2ε + 1) 2y2 + 3y + 9 where H H H = 3y . (35) ! 2 ε0 ε (y + 3) −y 1 − T(k) for k , 0, β This is to be compared with the well-known Onsager result K(k) =  ! H (28)  2y  − 1 for k = 0. (ε − 1)(2ε + 1) H  y + 3 = 3y. (36)  ε The details for solving Eq. (26) are given in Appendix D.  Equation (35) for the dielectric constant is a key result of One may be intrigued that the relation between F(k) and our theory. It predicts the dielectric constant of a polar liquid E0(k) takes different forms when k , 0 and k = 0. Mathe- H based on the magnitude of the permanent dipole momentµ ¯ matically, this difference is due to the singularity in T(k) at of the constituting molecule, the density of the fluid ρ, and Hk = 0. Physically, this feature has a deep origin rooted in the H the temperature T. When the system is sufficiently dilute, the existence of a monopole source for electric fields, which places expression reduces to ε = 1 + y to the first order in y, recovering a requirement on the value of T(k) at k = 0 so as to satisfy the the Debye theory for a dipolar gas. In Sec. III, we will apply electrostatic Poisson equation. Readers may refer to Ref. 61 H Eq. (35) to calculate the dielectric constants for a range of non- for a more in-depth discussion on this. polarizable liquid models, examine their agreement with the values obtained from simulation, and compare the prediction 2. The polarization and the dielectric constant of Eq. (35) to the results given by the Onsager equation and The polarization of a fluid under an electric field can previous field-theoretic approaches. be obtained by taking the derivative of the variational grand Note that one recovers Debye’s result for the dielectric potential with respect to the applied field using the statistical- constant ε = 1 + y if we take K(k) = 0. This is to be expected mechanical relation since setting K(k) = 0 amounts to ignoring the reaction field. It is interesting that Onsager’s resultH corresponds to K(k) = −1. δ βW H P(r) = − . (29) The reason for this is subtle and has to do with the equivalence δE r 0( ) between the applied field in our system and theH cavity field In the k-space, the above expression is equivalently written in Onsager’s construct, which we will elaborate further in the as following paragraph. 124108-6 B. Zhuang and Z.-G. Wang J. Chem. Phys. 149, 124108 (2018)

1 The equivalence between the applied field in our system The first term 2 µ · T(0) · µ represents the self-energy of the and the cavity field in Onsager’s construct can be seen by con- dipole, while the second term is due to the reaction field. Since sidering a macroscopically sized spherical system in an applied the free energy for the interaction between the dipole and the 1 1 2 17 field E0 (and the radius of the system can be taken to infin- reaction field is − 2 µ·R = − 2 βνµ , we identify the reaction ity eventually). A uniform polarization develops in the liquid field factor to be because of the applied field. If we carve out a molecular-sized dk 1 1 y cavity at the center of the liquid, as constructed by Onsager, ν = T(k) = , (42) (2π)3 β + ε a˜(k) 3ε b3 y + 1 the field E inside the cavity equals E because a spherical 0 0 c 0 H shell of uniform polarization does not exert a field at the cen- where we have introduced a microscopic cutoff b through ter.62 Because of this equivalence between the applied and the ∫ 1dk/(2π)3 = 1/b3, which defines the length scale of the self- cavity field, when K(k) = −1, the polarization is P = ε0yE0 interaction. Physically we expect b to be on the order of the 3 = ε yEc from Eq. (32). This means that, when we set molecular size; for a liquid, we may set b = 1/ρ. 0 H K(k) = −1 in our theory, the directing field—which exerts In Onsager’s treatment, the reaction field was computed a torque on the dipoles—is equal to the cavity field Ec. Since by placing a tagged dipole inside a spherical vacuum cavity H in Onsager’s theory, the cavity field is also the directing field of size b3 in a medium of dielectric constant ε. Using sim- (for a nonpolarizable liquid), setting K(k) = −1 in our theory ple electrostatics, the reaction field factor can be shown to be 1 2(ε−1) recovers Onsager’s result. ν = 3 . With our field-theoretic approach, the artifi- H 3ε0b 2ε+1 To further explore the connection between our theory and cial construct of a vacuum cavity is avoided. The resulting classical linear dielectric theory, we consider the case of a reaction field agrees with Onsager’s in the limit of strong uniform applied field. Using Eq. (32) for k = 0 and recalling dielectric (y & 3), but is weaker than Onsager’s for weak E0 = βE0, the polarization is dielectrics (y . 3). y(2y2 + 3y + 9) P = ε E . (37) 4. The free energy 0 (y + 3)2 0 It is of interest to consider the free energy in the linear By the connection between the cavity field and the electric response regime. The general expression can be obtained from 63 (Maxwell) field E in Onsager’s treatment, we have the grand potential in Eq. (16) combined with the variational (2ε + 1) conditions Eqs. (20a) and (20b). Here we consider the simple E = E . (38) 3ε 0 case of a uniform applied field E0. The total grand potential to quadratic order is On the other hand, for a linear dielectric medium, the polar- ! ! ization is related to the electric field in the material by 1 V 1 y 1 3y 2 βW = − ln + − βV ε0E − ρV, 2 b3 " y + 1 y + 1 # 2 y + 3 0 P = ε0(ε − 1)E. (39) (43) Combining the above three equations, we obtain our result for the dielectric constant, Eq. (35). where E0 = |E0| is the magnitude of the applied field and T R is evaluated based on Eq. (23), which gives 3. The reaction field β T = . (44) R 3 The concept of the reaction field, first introduced by 3ε0b (1 + y) Onsager,16 was a major advancement in our understanding of dielectric fluids. The concept is most easily illustrated by Performing the Legendre transform to the same order, we considering a tagged dipole in the fluid. The orientation of obtain the final Helmholtz free energy this dipole influences the orientation of other dipoles in its βF = βW + βµN surrounding, which then exerts a reaction field back on the ! Λ3 ! tagged dipole. This reaction field thus always points in the V 1 3y 2 N = ln(1 + y) − βV ε0E0 + N ln − N. same direction as the tagged dipole, and it can be formally 2b3 2 y + 3 V written as (45) R = βνµ, (40) Thus, the Helmholtz free energy due to the applied field where we have defined the reaction field factor ν. is We can determine the reaction field based on the effec- ! 1 3y 2 βF = − βV ε0E (46) tive single-particle potential in Eq. (17), which gives the E0 2 y + 3 0 free energy of a dipole in the absence of an applied field as which suggests that the free energy decreases when an electric field is applied. This is to be expected because the applied 1 f = µ · T · µ field induces a polarization in a direction that is energetically 0 2 R ! favorable. 1 1 dk β = µ · T(0) · µ − µ · T(k) · µ. We have thus developed the theoretical framework in 3 2 2 (2π) β + ε0a˜(k) this section. The framework includes a master variational H (41) grand potential for a general polar liquid, and from this grand 124108-7 B. Zhuang and Z.-G. Wang J. Chem. Phys. 149, 124108 (2018) potential, all other thermodynamic quantities can be derived. constants of the GAFF liquid models calculated using our the- By considering a weak applied field, we have derived an ory [Eq. (35)] are plotted against the corresponding simulation expression for the dielectric constant and this is a key result data as blue squares scattered on a log-log scale. Details of the of the theory. In Sec. III, we apply our expression for the parameters used in the calculation, and the numerical values dielectric constant to a range of nonpolarizable liquid mod- of the dielectric constants from both our theoretical predic- els and examine their agreement with the simulation-obtained tion and from the computer simulation, are provided in the values. supplementary material. A best-fit line is drawn through the scattered points, and the line has a slope of 1.02. In view of the highly coarse-grained nature of our theory, the agreement III. DIELECTRIC CONSTANTS OF MODEL LIQUIDS between the theory and the simulation is quite good. The actual A key result of our theory is the expression for the dielec- molecules in the simulation include detailed molecular struc- tric constant of a polar fluid presented in Eq. (35). This tures; some structures enhance dipole-dipole correlations (as expression computes the dielectric constant based on the num- in the case of some hydrogen bonding liquids), while others ber density and the permanent dipole moment of the fluid hinder dipole-dipole interactions. Such molecular details are molecules, as well as the temperature of the liquid. In this not captured in our coarse-grained theory. Therefore, it is not section, we apply the expression to calculate the dielectric surprising that the theory results in an underestimate of the constants of model liquids of small organic molecules and dielectric constants for some liquids and an overestimate for compare our results to the predictions of the Onsager equa- others. In Fig. 1(b), we similarly plot the theoretical vs. sim- tion and previous field-theoretic approaches. Because all real ulation dielectric constants for the OPLS/AA liquid models. molecules are polarizable but our current theory only considers Details of the parameters involved and the results for individ- permanent dipoles, it is more appropriate to use nonpolariz- ual molecules are also provided in the supplementary material. able liquid models as the basis for comparison. Among the For the OPLS/AA liquid models, the line of best-fit through major forcefields available today, two of the commonly used the theory vs. simulation values has a slope of 0.98, which forcefields for small molecules are the OPLS/AA (all-atom again demonstrates the overall good predictive power of our optimized potential for liquid simulations) and the GAFF theory. (generalized Amber force field) models. The dielectric con- As mentioned in the Introduction, there have been a num- stants for over a hundred molecules with these two forcefields ber of attempts in modeling the dielectric response in soft- have been computed by Caleman et al.53 using computer matter systems. The simplest approach is the self-consistent- simulation. To the best of our knowledge, Ref. 53 provides field theory (SCFT), which treats the field-based partition the most extensive set of dielectric constants obtained from function using the saddle-point approximation. Examples of computer simulation; therefore, we compare our predicted this approach include the ionic screening theory by Coalson dielectric constants with the corresponding values reported in and Duncan,43,44 the dipolar Poisson-Boltzmann equation by Ref. 53. Abrashkin et al.,45 the dipolar self-consistent-field theory by As the dielectric constants of the model liquids vary Nakamura et al.,46 and the generalized theory for polymers over two orders of magnitude, we plot the results on a containing dipoles on the backbone by Kumar et al.47 The logarithmic scale. All available simulation data reported in SCFT gives the following simple expression for the dielectric Ref. 53 are included for comparison. In Fig. 1(a), the dielectric constant:

FIG. 1. Dielectric constants of liquid models calculated using the theories εth vs. values obtained from simulations εsim plotted on a log-log scale. The liquid models are (a) the GAFF (generalized Amber force field) models and (b) the OPLS/AA (all-atom optimized potential for liquid simulations) models. The theories used included our dipolar renormalized gaussian fluctuation theory (DRGF, blue open squares), the Onsager equation (red open circles), the self-consistent field theory or the quadratic order field theory (SCFT/QOFT, green stars), and the Dipolar Poisson Boltzmann approach expanded to one-loop level (DPB-1Loop, orange open inverted triangles). Respectively, the lines of best fit for log εth vs. log εsim for the DRGF theory, the Onsager equation, the SCFT/QOFT theory, and the DPB-1Loop approach have slopes 1.02, 0.95, 0.82, and 1.40 for the GAFF models in (a) and 0.98, 0.92, 0.80, and 1.37 for the OPLS/AA models in (b). The black dashed line has a slope of 1 on the log-log scale, plotted for reference purposes. 124108-8 B. Zhuang and Z.-G. Wang J. Chem. Phys. 149, 124108 (2018)

βµ¯2 ρ to the size of the molecule. The cut-off length b is a fitting ε = 1 + . (47) 3ε0 parameter in the DPB approach, and we have found that the value of ε is very sensitive to the value of b used. To apply Since SCFT is a mean-field theory that ignores fluctua- the theory free of any adjustable parameters, we let b3 = 1/ρ tions, it is not surprising that the reaction field is not accounted so that b3 describes the average size of each molecule. The for in SCFT. Indeed, the SCFT result is the same as the Debye dielectric constant calculated using Eq. (49) is referred to as theory, which is applicable only when the molecules are suf- DPB-1Loop in the plots. In Figs. 1(a) and 1(b), we plot the ficiently dilute. In a dense liquid, the presence of the reaction DPB-1Loop dielectric constants against the simulation values field enhances dipole-dipole correlations; therefore, the SCFT for the GAFF and the OPLS/AA models, respectively. The is expected to underestimate the liquid dielectric constants. best-fit lines through the DPB-1Loop data points are found to In Figs. 1(a) and 1(b), we plot the SCFT dielectric constants have slopes 1.40 and 1.37 for the GAFF and the OPLS/AA against the simulation values for the GAFF and the OPLS/AA models, respectively, showing that the DPB-1Loop approach models, respectively. The best-fit lines through the data points significantly overestimates the dielectric constants of liquid are found to have slopes 0.82 and 0.80, respectively, confirm- models. ing our expectation that the SCFT indeed underestimates the In classical statistical-mechanical treatments of dielectric dielectric constants. liquids, a major improvement over Debye’s dilute-limit the- Very recently, Martin et al. presented a dielectric theory was provided by Onsager. As the Onsager theory involves ory for polarizable soft matter. In their work, a field-theoretic the same number of parameters as in our theory and is the approach is used to describe a system of polarizable “beads” best known analytical theory for dielectric constants, we com- and a simple gaussian approximation is employed to treat the pare the predictions from our theory with those from the field-based partition function. Since the field-theoretic action Onsager equation. In Figs. 1(a) and 1(b), we plot the dielectric is kept to the quadratic order in the work, we will call the theory constants predicted by the Onsager equation vs. the simu- a quadratic-order field theory (QOFT). The work focuses pri- lation values, where the lines of best-fit have slopes 0.95 marily on fluids of induced dipoles, but the authors have also and 0.92 for the GAFF and the OPLS/AA models, respec- provided the partition function for a system with only perma- tively. These are to be compared to the slopes of 1.02 and nent dipoles. By comparing Eqs. (10) and (14) in Ref. 50 to 0.98 by our theory. The level of agreement between theory second order in the field φ¯ in the exponent, we find that polar- and simulation is comparable between the Onsager equation izability α in the induced-dipole-only case is algebraically and our theory, but our theory produces slightly better agree- equivalent toµ ¯2/3 in the permanent-dipole-only case.64 There- ment. More importantly, by treating all particles equivalently, fore, the dielectric constant for the permanent-dipole-only case our theory provides a greater degree of self-consistency than given by the QOFT theory is [based on Eq. (44) in Ref. 50, Onsager’s approach, which requires the artificial construct converted to SI unit] of a cavity around a tagged molecule. One clear advantage βµ¯2 ρ of our theory is the ease with which to generalize to liquid ε = 1 + (48) 3ε0 mixtures. On closer examination of the variational approach that we which is, surprisingly, the same as the SCFT result. We are have developed in this work, we note that the inclusion of the unable to comment on the exact reason for the gaussian approx- variational parameter F is responsible for the improvement imation to reproduce the SCFT result, but we note that the over previous field-theoretic approaches. If we set the field F electric potential field fluctuates around zero in the gaus- to zero [i.e., setting K(k) = 0 in Eq. (28)], we recover the SCFT sian approximation and this may be insufficient for interac- result for the dielectric constant in Eq. (47). This suggests that tions beyond the saddle-point approximation to be taken into H in constructing the variational reference action L0 in Eq. (12), account. As the QOFT result is the same as the SCFT results, it is crucial to allow the field iG to fluctuate around a nonzero the comparison in Fig.1 for the SCFT applies equally to the average value F so that the effects of reaction field can be QOFT results. accounted for. Another field-theoretic dielectric theory developed in The effect of F may be further understood by looking recent years is the Dipolar Poisson-Boltzmann (DPB) 48,49 into the single-particle partition function. In the exact field- approach by Levy, Andelman, and Orland. In their work, theoretic action [Eq. (11)], the last term is the single-particle a system of ions and dipoles are considered and field-theoretic partition function under the auxiliary field iG methods are applied to derive the field-based grand partition function. The authors expand the free energy to first order in a · q = λ dr dΩ eiµ G(r), (50) loop expansion to obtain a closed form formula for the dielec- exact tric constant. For a system of only permanent dipole moments, whereas in the reference action that we proposed, the last term the dielectric constant is given by is in the form of the single-particle partition function under the ε2 4π field iG − F expanded to second order in the fields since ε = 1 + ε + 1 , (49) 1 3 1 + ε1 3ρb µ·[iG(r)−F(r)] qref = λ dr dΩ e 2 where ε1 = β ρµ¯ /3ε0 is the self-consistent-field contribu- 1 4πλ µ¯2 tion to the dielectric constant and b is a cut-off length in = constant + dr[iG(r) − F(r)]2 + ··· . (51) the momentum integration which is considered to be related 2 3 124108-9 B. Zhuang and Z.-G. Wang J. Chem. Phys. 149, 124108 (2018)

Here, the linear term vanishes due to the integral over the solid is important because the parameter space for mixtures—the angle of the dipole. Apparently, in the reference action, the different molecular species and different compositions—is too dipole is now in a “corrected” fluctuating field iG − F. This enormous to be representatively covered by computer sim- “corrected” field is in a similar spirit as Onsager’s approach, ulation; a simple theory is highly useful and desirable. In which established that only a portion of the local field acting addition, our theory can be extended to polarizable liquids, on a dipole exerts a torque on the dipole (the reaction field is where the polarizability and the van der Waals interactions always in the same direction as the dipole and, therefore, it does resulting from it have proved to be important.65 Very recently, not exert any torque). However, our theory is very different Grzetic, Delaney, and Fredrickson applied the QOFT frame- from Onsager’s in the approach for incorporating reaction- work to explore the connection between polarizability and field effects—while Onsager carefully examined the forces on the effective χ parameter in non-polar polarizable systems;66 a tagged molecule inside an artificially constructed vacuum our current work can be extended to polarizable systems and cavity, in our theory, the reaction field arises naturally from applied in a similar pursuit. We reserve these efforts for future properly treating the fluctuation-correlation in the entire fluid work. system. The past two decades have seen great success in the application of field theoretical methods to the study of a vast range of problems in the areas of soft matter, including poly- IV. CONCLUSIONS mers, liquid crystals, and ionic solutions.40,42,67 While earlier In this work, we have developed a coarse-grained theory work focused on the self-consistent mean-field approxima- for pure polar liquids. By introducing a gaussian reference tions,45,68–70 considerable progress has been made in under- action in the field-theoretic representation, we account for the standing the nontrivial and sometimes qualitative effects due electrostatic interactions through a self-consistent variational to fluctuations.51,71,72 The advent of field theoretical simu- procedure. Such a procedure allows us to treat all particles on lations41 has further expanded the capabilities of the field- the same footing and thus it is free of the artificial construct of theoretical methods and allowed some hitherto impossible a cavity, differing from Onsager’s dielectric theory which sin- phenomena to be described.73–76 In this work, we have shown gles out a particle in the liquid. Through the field-interaction that field-theoretical techniques can naturally capture the tensor and the average fluctuating field as variational param- nontrivial reaction field—another manifestation of fluctua- eters, our variational procedure mathematically renormalizes tion effects—in the dielectric response of molecular liquids. the dipole-dipole interaction strength and the local field act- In combination with field-theoretic representation of other ing on each dipole and, as a result, the procedure naturally degrees of freedom, the present theory contributes to the gen- accounts for the reaction field arising from the dipole-dipole eral field-theoretical coarse-grained modeling of soft-matter interactions. systems to allow the description of the dielectric response of A key result in our work is the expression for dielec- the materials. tric constant of a pure liquid, Eq. (35). This is a simple analytical expression that predicts the dielectric constant SUPPLEMENTARY MATERIAL with only two readily available parameters—the permanent dipole moment and the density of the molecules— The supplementary material contains the data and the requiring no adjustable parameters. For an extensive set of parameters used in Fig.1. nonpolarizable model liquids, we have computed the dielectric constants with our theory, and the best-fit line through ACKNOWLEDGMENTS the theoretically predicted dielectric constants vs. the corresponding simulation-observed values has a slope very close We thank Issei Nakamura, Rui Wang, Kevin Shen, Jian to 1 on a log-log plot. We have also compared the predictions Jiang, Nayef Alsaifi, and Pengfei Zhang for helpful discus- of our theory to earlier dielectric theories derived using other sions. B.Z. gratefully acknowledges the support by an A-STAR field-theoretic approximations and the results have shown that fellowship. Acknowledgement is also made to the donors of our theory produces better agreement with the simulation the American Chemical Society Petroleum Research Fund for results. partial support of this research. We have applied the variational theory to homogeneous pure polar fluids in a weak applied field, where the polar- APPENDIX A: IDENTITY TRANSFORMATION ization responds linearly to the applied field. Beyond this, OF THE GRAND PARTITION FUNCTION the full grand potential Eq. (16), along with the two constitutive relations, Eqs. (20a) and (20b), is applicable to sys- Since the dipole-dipole interaction tensor T(r) does tems in the presence of strong and/or nonuniform applied not have an inverse operator, we perform the identity fields. Therefore, the theory can be readily applied, for exam- transformations using the Faddeev-Popov method40,42 by ple, to study dipolar fluid in the vicinity of strongly charged introducing the δ-functional into the partition function in ions. Eq. (10), Since the derivation of our theory does not require sin- ˆ gling out a special tagged molecule or the use of a cavity 1 = DP δ[P(r) − Pˆ (r)] = DP DG ei ∫ dr G(r)·[P(r)−P(r)], construct as in the theory by Onsager, it is straightforward to extend the DRGF theory to liquid mixtures. Such an extension (A1) 124108-10 B. Zhuang and Z.-G. Wang J. Chem. Phys. 149, 124108 (2018)

− 1 dr dr0P(r)T(r−r0)P(r0) where δ[f(r)] is the generalization of the multivariate δ- Z(N) = DP DG e 2 ∫ ∫ function. With the δ-functional in Eq. (A1), we transform the Boltzmann factor as × e−i ∫ dr P(r)·G(r)+∫ dr P(r)·E0(r) −βU − 1 ∫ dr ∫ dr0P(r)T(r−r0)P(r0) e = DP DG e 2 ( )N 1 × dr dΩ eiµ·G(r) . (A3) ˆ 3 × e−i ∫ dr P(r)·G(r)+∫ dr P(r)·E0(r)+i ∫ dr P(r)·G(r), (A2) 4πΛ where only the last term in the exponential depends explicitly Substituting Eq. (A3) into Eq. (5) and summing over all N on the instantaneous molecular configuration. Use of Eq. (A2) leads to the field-based grand partition function and the field- in the Boltzmann factor in Z(N) in Eq. (6) leads to theoretic action given by Eqs. (10) and (11).

APPENDIX B: EVALUATION OF THE VARIATIONAL BOUND In this section, we present the evaluation of the variational grand potential W. The evaluation involves the operator Q = T + A and its inverse Q−1. We start by deriving an expression for Q−1 in the Fourier space −1 1 Q (k) = [T(k) + A(k)]−1 = [T(k) +a ˜(k)1]−1 = [1 +a ˜(k)−1T(k)]−1 a˜(k) H H1 H H H = [1 − a˜(k)−1T(k) +a ˜(k)−2T(k)2 − a˜(k)−3T(k)3 + ··· ] a˜(k) ! !2 !3 1 β H kk Hβ kk Hβ kk = 1 − + − + ··· 2 2 2 a˜(k)  ε0a˜(k) k ε0a˜(k) k ε0a˜(k) k    1  β 1 kk 1 ε  = 1 − = 1 − 0 T(k) . (B1) β 2 a˜(k)  ε0a˜(k) k  a˜(k) " β + ε0a˜(k) #  1 + ε a˜(k)   0  H    

In the fourth equality above, we have used the fact that kk/k2 where the last equality is due to the chain rule and the : symbol is a projection operator and, thus, (kk/k2)n = kk/k2 for any indicates scalar product of two tensors. Using the fact that det positive integer n. Γ(θ) can be written as a gaussian functional integral as in Since the reference action L0 is gaussian with respect to Eq. (B3), we have both P and G, we can evaluate Ξ0 using standard techniques for gaussian integrals. The result is δ ln det Γ(θ) −1 ! 1 δΓ (k; θ) det A 2 0 1 −1 0 0 0 ∫ dr ∫ dr [E0(r)−F(r)]Q (r−r )[E0(r )−F(r )] 0 −1 Ξ0 = e 2 , − 1 ∫ dk ξ(k0)Γ (k0;θ)ξ(−k0) T A H 1 D − 2 (2π)3 det( + ) (2π)3 ∫ ξ ξ(k)ξ( k)e H H H = 0 −1 (B2) H −H1 ∫ dk ξ(k0)Γ (k0;θ)ξ(−k0) ∫ Dξ e 2 (2π)3 where the determinant of a tensor M is given in terms of a H H V H = −Γ(k, θ)δ(k = 0) = −Γ(k, θ) , (B5) functional integral as (2π)3 H H − 1 − 1 dr dr0 ξ(r)M(r−r0)ξ(r0) (det M) 2 = Dξ e 2 ∫ ∫ where, in the last equality, we have used the following interpretation of δ(k = 0):78 − 1 ∫ dk ξ(k)M(k)ξ(−k) = Dξ e 2 (2π)3 . (B3) H H H 1 1 V δ(k = 0) = dr eik·r = dr 1 = . 3 k=0 3 3 The factor det A/det(T + A) can be evaluated using the (2π) (2π) (2π) “charging” method described in Appendix B of Ref. 77. Let (B6) us deﬁne Γ(θ) = θT + A. Then, − ! Furthermore, an expression for Γ 1 can be derived using a det A ln = ln det Γ(θ = 1) − ln det Γ(θ = 0) similar method that is used in Eq. (B1), det(T + A) Γ(θ=1) δ ln det Γ(θ) −1 −1 1 ε0θ = dk : δΓ (k; θ), Γ (k; θ) = 1 − T(k) . (B7) −1 a˜(k) βθ + ε a˜(k) Γ(θ=0) δΓ (k; θ) " 0 # H H H H (B4) Then, substituting Eqs. (B5) and (B7) into (B4), we have 124108-11 B. Zhuang and Z.-G. Wang J. Chem. Phys. 149, 124108 (2018)

! −1 det A dk 1 dΓ (k; θ) ln = Γ(k; θ): dθ det(T + A) (2π)3 dθ θ=0 H H dk 1 ε2 = θT(k) + A(k) : 0 T(k)dθ π 3 θ β ε a k 2 (2 ) θ=0 f g ( + 0 ˜( )) H H H ! dk ε dk ε a˜(k) β = V 0 trT(k) − V ln 0 + . (B8) 3 3 (2π) β + ε0a˜(k) (2π) " β + ε0a˜(k) β + ε0a˜(k) # H Next, we evaluate hL − L0i0, which can be further written as D E 1 hL − L i = −λ dr dΩ eiµ·G(r) − drh[G(r) + iF(r)]A−1(r − r0)[G(r0) + iF(r0)]i . (B9) 0 0 0 2 0 iµ·G(r) The quantity he i0 can be evaluated using standard techniques for gaussian integrals. The result is D iµ·G(r)E − 1 µ·T ·µ+γ(r)·µ e = e 2 R . (B10) 0 −1 0 0 0 The average h[G(r) + iF(r)]A (r − r )[G(r ) + iF(r )]i0 can be evaluated using standard techniques for gaussian integrals as well. For simplicity, we write ξ(r) = G(r) + iF(r). Then, D E 1 ξ(r)A−1(r − r0)ξ(r0) = DP DG ξ(r)A−1(r − r0)ξ(r0)e−L0[P,G] 0 Ξ0 ! 1 δ ∂ 0 −L0[P,G]+i ∫ dr1J(r1)·ξ(r1) = · A(r − r ) · 0 DP Dξ e . (B11) Ξ iδJ(r) i∂J(r ) J=0 0 Evaluation of the above gaussian integral gives

D E dk ε dk 1 dr dr0 ξ(r)A−1(r − r0)ξ(r0) = V 0 trT(k) − E (k) − F(k) 0 π 3 β + ε a˜(k) π 3 a˜(k) 0 (2 ) 0 (2 ) f g !H2 H H ε × 1 − 0 T(k) E (−k) − F(−k) . (B12) β + ε a˜(k) 0 0 f g H H H The variational grand potential in Eq. (16) is obtained upon substitution of the relevant terms into Eq. (13) followed by additional algebraic simpliﬁcation.

APPENDIX C: STATIONARY POINT OF THE VARIATIONAL GRAND POTENTIAL In this section, we extremize βW in Eq. (16) with respect to the variational parameters a˜(k) and F(k). First, setting δ βW/δa˜(k) = 0, we obtain H !3 1 β2 ε V − 0 E (k) − F(k) T(k) E (−k) − F(−k) 2 2 β + ε a˜(k) 0 0 a˜(k) β + ε0a˜(k) 0 f g f g H H H H H !2 βε ε = λ dr dΩ 0 µ · T(k) · µ − 0 µ · T(k) · [E (k) − F(k)]eik·r e−f (µ,r). (C1) 2(β + ε a˜(k))2 β + ε a˜(k) 0  0 0   H H H H   

Next, setting δ βW/δF(k) = 0, we obtain APPENDIX D: EVALUATION OF F(k) IN THE LINEAR RESPONSE REGIME ! ε H H 0 T(k) E (k) − F(k) Here, we solve Eq. (26) for the variational parameter F(k). β + ε a˜(k) 0 0 f g For k , 0, the matrix T(k) is well defined, and therefore, we H H H H = λ dr dΩ T(k)µe−f (µ,r)−ik·r. (C2) can express F(k) as H !−1 ! H H ε0 ε0 F(k) = −y 1 + y T(k) 1 − T(k) E (k). (D1) Substituting Eq. (C2) into Eq. (C1), we simplify Eq. (C1) β β 0 to Eq. (20a). In addition, based on the uniqueness the- H H H H ε0 orem in electrostatic interactions, Eq. (C2) simplifies to The inverse operator of 1 + y β T(k) can be found using a Eq. (20b). procedure similar to that presented in Eq. (B1). The procedure H 124108-12 B. Zhuang and Z.-G. Wang J. Chem. Phys. 149, 124108 (2018) leads to 18J. G. Kirkwood, J. Chem. Phys. 7, 911 (1939). 19 !−1 M. Neumann, Mol. Phys. 50, 841 (1983). ε y ε 20 1 + y 0 T(k) = 1 − 0 T(k). (D2) J. N. Wilson, Chem. Rev. 25, 377 (1939). β 1 + y β 21D. V. Matyushov and B. M. Ladanyi, J. Chem. Phys. 110, 994 (1999). 22 H H A. Tani, D. Henderson, J. A. Barker, and C. E. Hecht, Mol. Phys. 48, 863 Then, the evaluation of Eq. (D2) gives the expression for F(k) (1983). at k 0, 23S. Goldman and C. Joslin, J. Chem. Phys. 99, 3021 (1993). , 24 ! H D. W. Jepsen, J. Chem. Phys. 45, 709 (1966). ε0 25D. Chandler, J. Chem. Phys. 67, 1113 (1977). F(k) = −y 1 − T(k) E0(k) for k , 0. (D3) 26 β G. Nienhuis and J. M. Deutch, J. Chem. Phys. 55, 4213 (1971). 27 H H H J. S. Høye and G. Stell, J. Chem. Phys. 61, 562 (1974). For k = 0, T(k) is not well-defined in the k-space. The 28B. U. Felderhof, G. W. Ford, and E. G. D. Cohen, J. Stat. Phys. 28, 135 expression in Eq. (26) is best solved in the position space (1982). H 29M. S. Wertheim, J. Chem. Phys. 55, 4291 (1971). instead. We obtain the equation in the position space by inverse 30M. S. Wertheim, Mol. Phys. 25, 211 (1973). Fourier transform of Eq. (26), which gives 31S. A. Adelman and J. M. 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